tag:blogger.com,1999:blog-5730391.post1148309150613528159..comments2019-01-17T13:16:39.713-08:00Comments on Blaine's Puzzle Blog: NPR Sunday Puzzle (Mar 3, 2013): Dinner Party Musical ChairsBlainehttp://www.blogger.com/profile/06379274325110866036noreply@blogger.comBlogger157125tag:blogger.com,1999:blog-5730391.post-84244955094643329032013-03-10T01:51:23.414-08:002013-03-10T01:51:23.414-08:00I've made my submission. I'm I happy with...I've made my submission. I'm I happy with it?<br /><br />I happen to know a certain king would be <b><i>unhappy</i></b> with <b><i>one</i></b> of the three-word sayings I submitted. He even proposed an <b><i>improved</i></b> three-word phrase to replace it, but unfortunately his improvement fails the criteria of the puzzle.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-5730391.post-53140115145776288392013-03-09T23:03:41.764-08:002013-03-09T23:03:41.764-08:00New puzzle is up. The key here is to take your ti...New puzzle is up. The key here is to take your time. If that doesn!t work, then just force it.AbqGuerrillahttps://www.blogger.com/profile/08439641600563812511noreply@blogger.comtag:blogger.com,1999:blog-5730391.post-29696662486146960102013-03-09T22:52:11.470-08:002013-03-09T22:52:11.470-08:00The new puzzle came up a bit ago.
Next week's...The new puzzle came up a bit ago.<br /><br />Next week's challenge: Think of two familiar three-word sayings in which all three words are the same length. The middle word in both sayings is the same. In each saying, the first and last words rhyme with each other. What two sayings are these?<br /><br />I already have an answer and am wondering if it the same as expected. Who thought it would be up so soon? Sorry I have not come up with a hint yet that is not too revealing, so you are just going to have to waitskydiveboyhttps://www.blogger.com/profile/17174073226290431753noreply@blogger.comtag:blogger.com,1999:blog-5730391.post-18562600926335020352013-03-09T19:47:06.892-08:002013-03-09T19:47:06.892-08:00Enter man in a bejeweled dinner jacket. The math s...Enter man in a bejeweled dinner jacket. The math symposium director looks at him and says "Sequence, sequence...not sequins."<br /><br />Ok, on to the next puzzle, please.Word Womanhttps://www.blogger.com/profile/15491300694641304112noreply@blogger.comtag:blogger.com,1999:blog-5730391.post-43208983609456864672013-03-09T17:04:37.859-08:002013-03-09T17:04:37.859-08:00This comment has been removed by the author.Word Womanhttps://www.blogger.com/profile/15491300694641304112noreply@blogger.comtag:blogger.com,1999:blog-5730391.post-47899924372767589772013-03-09T16:16:35.656-08:002013-03-09T16:16:35.656-08:00Just for the record.
Just for the record.<br />Paulhttps://www.blogger.com/profile/11114786604125384958noreply@blogger.comtag:blogger.com,1999:blog-5730391.post-60696943557344140352013-03-09T16:10:43.888-08:002013-03-09T16:10:43.888-08:00Paul, we did both. Measured the height & widt...Paul, we did both. Measured the height & width of the kids' faces and averaged the ratios to 1.61. Had us jumping out of our chairs and onto our phi-t. <br /><br />Journey...not the destination ;-). That's what I wrote on my SATs. <br />All this OEIS talk has me wondering if mathematicians, like prisoners and jokes, walk around saying A82031 and A3424 to each other at math conferences. . .Word Womanhttps://www.blogger.com/profile/15491300694641304112noreply@blogger.comtag:blogger.com,1999:blog-5730391.post-25484471978268500702013-03-09T14:42:03.146-08:002013-03-09T14:42:03.146-08:00And I thought you were playing Fibonacci chairs ye...And I thought you were playing Fibonacci chairs yesterday. Just goes to show!<br />Uh...the 'charm' of mathematics is it's 'certainty' ???????????Paulhttps://www.blogger.com/profile/11114786604125384958noreply@blogger.comtag:blogger.com,1999:blog-5730391.post-14929609823850266912013-03-09T14:28:14.601-08:002013-03-09T14:28:14.601-08:00@AbqGuerrilla, glad you enjoy botanical humor ;-)....@AbqGuerrilla, glad you enjoy botanical humor ;-). As to the rope and stool, it makes me think of Whodunnit books...Although in Utah maybe they are Hoodoo it books?<br /> As to the uncertainty of the answers for the math puzzles, that's part of their charm for me. That, and all the good stuff I learn along the way.<br /> We talked about phi yesterday with Kg crowd.Several kids thought I was saying feet! <br /> Another weekend snowstorm here so plenty of time for puzzle solving.Word Womanhttps://www.blogger.com/profile/15491300694641304112noreply@blogger.comtag:blogger.com,1999:blog-5730391.post-55719440161321302782013-03-09T13:37:52.670-08:002013-03-09T13:37:52.670-08:00You can fool all of the people some of the time, a...You can fool all of the people some of the time, and some of the people all of the time, but you can't fool all of the people all of the time.Paulhttps://www.blogger.com/profile/11114786604125384958noreply@blogger.comtag:blogger.com,1999:blog-5730391.post-32476784812264472902013-03-09T13:10:31.299-08:002013-03-09T13:10:31.299-08:00Hola WW ~ I enjoy the college basketball classic a...Hola WW ~ I enjoy the college basketball classic almost as much as your botanical humor. As for my absence this week, the math puzzles just don't do it for me. I am quite good at solving them, but the fact that you never really know for sure if you have the correct answer always leaves me feeling like it's an exercise in futility. I thought about purchasing a math puzzle book on Amazon last month and when I clicked on it, a pop-up window surfaced with the message: "People who bought this book also bought a rope and a stool." That kind of sums it up for me. "See" you manana, no doubt.AbqGuerrillahttps://www.blogger.com/profile/08439641600563812511noreply@blogger.comtag:blogger.com,1999:blog-5730391.post-10968793169236036032013-03-09T11:44:33.670-08:002013-03-09T11:44:33.670-08:00Congratulations, Laura Leonard.Congratulations, Laura Leonard.Paulhttps://www.blogger.com/profile/11114786604125384958noreply@blogger.comtag:blogger.com,1999:blog-5730391.post-15660354139539888812013-03-09T04:47:59.248-08:002013-03-09T04:47:59.248-08:00Continuing from my last post, I now want to show w...Continuing from my last post, I now want to show why<br /><br />sigma(n) = Fib(n-1) + Fib(n+1) + 2<br /><br />where Fib(n) is the nth term of the Fibonacci series (i.e., 1,1,2,3,5,8, ...). The proof is by induction on m, m > 2.<br /><br />As others have show above, sigma(3)=6, and Fib(2)=1 and Fib(4) = 3, so<br /><br />sigma(3) = Fib(2) + Fib(4) + 2 = 1 + 3 + 2 = 6.<br /><br />Now, assume<br /><br />sigma(m) = Fib(m-1) + Fib(m+1) + 2<br /><br />We must show that<br /><br />sigma(m+1) = Fib(m) + Fib(m+2) + 2<br /><br />From my previous post<br /><br />sigma(m) = sigma(m-2) + sigma(m-1) - 2<br /><br />So,<br /><br />sigma(m+1) = sigma(m-1) + sigma(m) - 2<br /><br />Substituting the induction hypothesis,<br /><br />sigma(m+1) = Fib(m-2) + Fib(m) + 2 + Fib(m-1) + Fib(m+1) + 2 - 2<br /><br />or<br /><br />sigma(m+1) = Fib(m-2) + Fib(m-1) + Fib(m) + Fib(m+1) + 2<br /><br /><br />By the definition of the Fibonacci series,<br /><br />Fib(m) = Fib(m-2) + Fib(m-1), and<br />Fib(m+2) = Fib(m) + Fib(m+1)<br /><br />Substituting this gives<br /><br />sigma(m+1) = Fib(m) + Fib(m+2) + 2<br /><br />which is the desired result, and we have proved<br /><br />sigma(n) = Fib(n-1) + Fib(n+1) + 2<br /><br />by induction.<br /><br />We can now apply the closed form solution based on the Fibonacci series explained previously.actonbellhttps://www.blogger.com/profile/05344781460971466611noreply@blogger.comtag:blogger.com,1999:blog-5730391.post-34016323074716547642013-03-09T04:23:46.454-08:002013-03-09T04:23:46.454-08:00All week I've wanted a mathematical/logical ex...All week I've wanted a mathematical/logical explanation from basic principles for why the Fibonacci series can be used to find the general answer to this problem (i.e., not just looking at the OEIS and finding a series that seems to fit this case). I think I have found such an explanation. It's fair long, so I'll give it in 2 posts.<br /><br />First I want to show that<br /><br />sigma(n) = sigma(n-2) + sigma(n-1) - 2<br /><br />where sigma(n) is the answer to this problem for a table size of n. Define a function swap(n) that is the number of 2-seat swaps for a table of n, INCLUDING 0 swaps. Then,<br /><br />sigma(n) = swap(n) + 2<br /><br />where the 2 accounts for the cases where everyone moves right or left.<br /><br />Now, for a table size of n, focus on any pair of adjacent seats. There are exactly 5 all-inclusive cases:<br /><br />1. Both seats do not swap.<br />2. The leftmost seat swaps with the seat to the left and the rightmost does not swap;<br />3. The rightmost seat swaps with the seat to the right and the leftmost does not swap.<br />4. The seats swap with each other.<br />5. The leftmost seat swaps with the seat to the left and the rightmost seat swaps with the seat to the right.<br /><br />Now, consider a table size of (n-1), and focus on any seat. There exactly 3 cases:<br /><br />A. The seat does not swap.<br />B. The seat swaps with the seat to the left.<br />C. The seat swaps with the seat to the right.<br /><br />Case A maps exactly to Case 1 for the table size of n. By "maps" I mean that the remaining (n-2) seats have exactly the same possible arrangements. Case B maps to Case 2, and Case C maps to case 3.<br /><br />Now, consider a table size of (n-2) and focus on the space between any pair of seats. There are exactly 2 case:<br /><br />a. The seats on either side do not swap.<br />b. The seats on either side do swap.<br /><br />Case a maps to case 4 for a table size of n, and case b maps to case b maps to case 5.<br /><br />Therefore,<br /><br />swaps(n) = swaps(n-2) + swaps(n-1)<br /><br />but<br /><br />sigma(n) = swaps(n) + 2 or swaps(n) = sigma(n) - 2)<br /><br />So, substituting,<br /><br />sigma(n) = (sigma(n-2) - 2) + (sigma(n-1) - 2) + 2<br /> = sigma(n-2) + siga(n-1) - 2<br />actonbellhttps://www.blogger.com/profile/05344781460971466611noreply@blogger.comtag:blogger.com,1999:blog-5730391.post-33375453237860308502013-03-08T16:51:37.002-08:002013-03-08T16:51:37.002-08:00Did anyone from Blaine's Blog get the call?Did anyone from Blaine's Blog get the call?Word Womanhttps://www.blogger.com/profile/15491300694641304112noreply@blogger.comtag:blogger.com,1999:blog-5730391.post-87598811797943861622013-03-08T14:05:26.876-08:002013-03-08T14:05:26.876-08:00Ouch! Snipper and Blaine, one and the same?Ouch! Snipper and Blaine, one and the same?zeke creekhttps://www.blogger.com/profile/12559686966843380823noreply@blogger.comtag:blogger.com,1999:blog-5730391.post-28344952285869353912013-03-08T06:40:57.102-08:002013-03-08T06:40:57.102-08:00Zeke Creek, I'm surprised you weren't tuto...Zeke Creek, I'm surprised you weren't tutored by Blaine. ;-)Word Womanhttps://www.blogger.com/profile/15491300694641304112noreply@blogger.comtag:blogger.com,1999:blog-5730391.post-70796671049377210732013-03-08T05:39:39.762-08:002013-03-08T05:39:39.762-08:00In a slightly different question, where everybody ...In a slightly different question, where everybody had to move one to the left or the right, there would be only four possible re-seatings.Elliott Linehttps://www.blogger.com/profile/08698916440262184168noreply@blogger.comtag:blogger.com,1999:blog-5730391.post-60770252957219090402013-03-08T05:00:11.679-08:002013-03-08T05:00:11.679-08:00I guessed 52.I guessed 52.Uncle Johnhttps://www.blogger.com/profile/13769784532885433156noreply@blogger.comtag:blogger.com,1999:blog-5730391.post-52298737682173791172013-03-08T00:43:56.739-08:002013-03-08T00:43:56.739-08:00My answer was 56. I didn't account for the ove...My answer was 56. I didn't account for the overlap. Using my superficial knowledge of algebra I went with a combination equation.<br />@ ww halfway between mine and yours was the right answer. 1/2 point apiece.<br />Remember your towel and don't panic.<br />Zeekphodzeke creekhttps://www.blogger.com/profile/12559686966843380823noreply@blogger.comtag:blogger.com,1999:blog-5730391.post-69941075223395010732013-03-07T22:07:21.445-08:002013-03-07T22:07:21.445-08:00Domani never comes.Domani never comes.skydiveboyhttps://www.blogger.com/profile/17174073226290431753noreply@blogger.comtag:blogger.com,1999:blog-5730391.post-45473908823182297272013-03-07T21:01:15.881-08:002013-03-07T21:01:15.881-08:00I think it may just be partly the way math was gen...I think it may just be partly the way math was generally taught when we were learning. It was so esoteric and theoretical. Lots of workbooks. <br /><br />My kids had lots of manipulatives (beans or m and m's work just fine) so numbers had a tangible kinesthetic meaning to them. Once they could feel numbers it made sense in their developing brains.<br /><br />It makes sense to me that you liked geometry most because it has that tangible (and tangential;-)) feel.<br /><br />I wish you could experience these kindergarteners. There are 8 in the class. I placed several piles of beans on the table and asked them how many beans in the next pile? I gave them the clue of "Look what happens when we push together this pile of 1 and 1, now 1 and 2." Every kid placed the piles in sequence 1,1,2,3,5,8,13, and 21. One kid went up to 89!<br /><br /> At the end of class, Isobel said "And we're even a Fibonacci number class!"<br /><br /> Tomorrow can't come soon enough...<br /> Word Womanhttps://www.blogger.com/profile/15491300694641304112noreply@blogger.comtag:blogger.com,1999:blog-5730391.post-47569800220572048522013-03-07T20:47:56.370-08:002013-03-07T20:47:56.370-08:00Excellent. Thank you.Excellent. Thank you.PlannedChaoshttps://www.blogger.com/profile/12366856086658014260noreply@blogger.comtag:blogger.com,1999:blog-5730391.post-56320378247110602662013-03-07T20:38:36.028-08:002013-03-07T20:38:36.028-08:00Planned Chaos, If you do the power series expansio...Planned Chaos, If you do the power series expansion of the indicated quotient in powers of x, the coefficients of the successive powers of x<br />are the successive terms in the sequence. If the sequence you were trying to represent were all ones, you would write 1/(1 - x). The power series <br />expansion would be 1 + x + x^2 + x^3 + ..., and<br />all the coefficients are 1.EKWhttps://www.blogger.com/profile/07620788732511183218noreply@blogger.comtag:blogger.com,1999:blog-5730391.post-62586693965981852162013-03-07T20:16:02.555-08:002013-03-07T20:16:02.555-08:00I really cannot comment on Enya's description ...I really cannot comment on Enya's description as I am not proficient at math, but at least his is the most understandable to me. I was a terrible student in school and learned mostly by osmosis and the fact that I read adult books instead of paying attention or studying. I have to say that I loved geometry, much to my surprise, as it is puzzles. I have never been fully able to understand why I was a lousy student, but I have a high IQ and love to learn when it is intelligently provided. I have little good to say of my school teachers. <br />I never doubted that I was able to solve this puzzle, but I had things to do and was not excited about the tedium of listing the possibilities, which I believed were more than they actually are. Therefor I took a pass on this one and perhaps now regret it. You can just chalk it up to my being lazy this time.skydiveboyhttps://www.blogger.com/profile/17174073226290431753noreply@blogger.com